Riemannian Geometry Course | NPTEL | Prof. Ved Vivek Datar IISc
Course Details
| Exam Registration | 33 |
|---|---|
| Course Status | Ongoing |
| Course Type | Elective |
| Language | English |
| Duration | 12 weeks |
| Categories | Mathematics |
| Credit Points | 3 |
| Level | Postgraduate |
| Start Date | 19 Jan 2026 |
| End Date | 10 Apr 2026 |
| Enrollment Ends | 02 Feb 2026 |
| Exam Registration Ends | 20 Feb 2026 |
| Exam Date | 24 Apr 2026 IST |
| NCrF Level | 4.5 — 8.0 |
Master the Language of Curved Spaces: A Deep Dive into Riemannian Geometry
Welcome to a comprehensive exploration of one of the most beautiful and fundamental branches of modern mathematics. This 12-week postgraduate course on Riemannian Geometry, offered by the esteemed Prof. Ved Vivek Datar from the Indian Institute of Science (IISc) Bangalore, is designed to provide a rigorous foundation in the study of curved spaces. Riemannian geometry is the mathematical framework underlying Einstein's theory of General Relativity and is indispensable in fields ranging from geometric analysis and topology to computer graphics and machine learning.
About the Instructor: Prof. Ved Vivek Datar
Prof. Ved Vivek Datar is a leading expert in geometric analysis and complex differential geometry at IISc Bangalore. His research focuses on understanding the deep connections between the existence of canonical metrics, solutions to natural partial differential equations on complex manifolds, and the topological or algebraic-geometric constraints of these spaces. His work in Kähler geometry involves tackling comparison and rigidity problems, exploring how curvature conditions shape the very fabric of complex manifolds. This course is born from his expertise and passion for the subject, aiming to build a strong conceptual and technical foundation for aspiring mathematicians.
Course Overview and Objectives
This is a rigorous, postgraduate-level course that begins with a concise review of essential prerequisites like smooth manifolds and the geometry of curves and surfaces. It then systematically introduces the core concepts of Riemannian geometry.
The primary aim of this course is to fill a significant gap in the differential geometry offerings on the NPTEL platform. It will equip students with the necessary toolkit to branch out into advanced topics like geometric analysis, complex geometry, and geometric topology. Prof. Datar also plans to offer a follow-up course in geometric analysis, for which this course will serve as a crucial prerequisite.
Who Should Enroll?
INTENDED AUDIENCE:
- Masters and PhD students in Mathematics.
- Advanced undergraduate students with a strong mathematics background.
- Researchers in physics (especially relativity) and related fields seeking a solid mathematical foundation.
PREREQUISITES:
- Must Have: A firm grasp of multivariable calculus and smooth manifolds. Students must be comfortable with calculus on manifolds, tensors, differential forms, and Stokes’ theorem.
- Helpful but Not Required: A prior course on curves and surfaces. The course will be designed to be independent of such a background.
Detailed 12-Week Course Layout
| Week | Topics Covered |
|---|---|
| Week 1 | Introduction, review of curves and surfaces, Review of smooth manifolds – tensors, differential forms, Stokes’ theorem |
| Week 2 | Introduction to Riemannian metrics, examples and basic constructions |
| Week 3 | Levi-Civita connection, induced connection on tensors and forms, Parallel transport |
| Week 4 | Curvature of Levi-Civita connection, Sectional curvature, Ricci curvature and scalar curvature, geometry of sub-manifolds |
| Week 5 | Geodesics, first variation formula |
| Week 6 | Local behaviour of geodesics, exponential map, normal coordinates |
| Week 7 | Metric geometry, Hopf-Rinow theorem, regularity of distance function |
| Week 8 | 2nd variation formula, Jacobi fields, index form |
| Week 9 | Jacobi fields (continued), characterization of space forms |
| Week 10 | Catch-up week. If on schedule, introduction to comparison geometry. |
| Week 11 | Overview of comparison geometry (Rauch, Myers and Bishop-Gromov theorems) |
| Week 12 | Introduction to the Bochner technique |
Recommended Textbooks and Resources
To supplement the lectures, students are encouraged to refer to the following classic texts and the instructor's own notes:
- Gallot, Hulin, Lafontaine: Riemannian Geometry (3rd Ed., Springer). A comprehensive and detailed text.
- Peter Petersen: Riemannian Geometry (Graduate Texts in Mathematics, Springer). Known for its clarity and modern approach.
- John M. Lee: Riemannian Geometry: An Introduction to Curvature (Graduate Texts in Mathematics, Springer). An excellent first reading for the subject.
- Ved Datar: Lectures on Riemannian Geometry (Online notes available on the instructor's webpage). Directly aligned with the course content.
Why Study Riemannian Geometry?
Riemannian geometry is more than just an abstract mathematical discipline. It provides the vocabulary and tools to quantify and understand curvature, distance, and shape in higher dimensions. From describing the curvature of the universe in cosmology to optimizing paths in data science (a field known as geometric deep learning), its applications are vast and growing. This course will not only teach you the theorems but will also develop the geometric intuition needed to visualize and work with curved spaces.
Join Prof. Ved Vivek Datar on this 12-week intellectual journey to unravel the mysteries of shape, curvature, and space. Enroll in this NPTEL course to build a powerful foundation that will open doors to cutting-edge research in mathematics and theoretical physics.
Enroll Now →